\(X\)-elliptic operators and \(X\)-control distances
type: Item
instance of: scholarly article
title: \(X\)-elliptic operators and \(X\)-control distances
zbMATH Open document ID: 1029.35102
author: Ermanno Lanconelli
author: Alessia E. Kogoj
published in: Ricerche di Matematica
publication date: 2003-04-22T00:00:00Z
review text: The paper deals with linear second-order operator \(L\) in divergence form with nonnegative characteristic form. \(L\) is assumed to be \(X\)-elliptic. It means the existence of a family \(X=(X_1,\dots ,X_m)\) of locally Lipschitz continuous vector fields on \(\mathbb{R}^N\) such that, for a suitable constant \(c>0\), \[ \frac{1}{c}{\sum_{j=1}^m} \langle X_j(x),\xi_j\rangle^2\leq {\sum_{j=1}^m} a_{ij}(x)\xi_i\xi_j\leq c {\sum_{j=1}^m}\langle X_j(x),\xi_j\rangle^2 \] for every \(x,\xi\in\mathbb{R}^N\). A general condition on \(X\) assuring a scale invariant Harnack inequality for the positive weak solutions to \(Lu=0\) is established.
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35M10
Mathematics Subject Classification ID: 35K65
zbMATH DE Number: 1898324
zbMATH Keywords: \(X\)-elliptic operators
zbMATH Keywords: weak solution
zbMATH Keywords: Harnack inequality
zbMATH Keywords: degenerate elliptic equations
zbMATH Keywords: locally Lipschitz continuous vector fields
reviewed by: G. Jaiani
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1898324